Green's Function Solution of the Semi-Space with Double Shallow-Buried Inclusions

2011 ◽  
Vol 462-463 ◽  
pp. 518-523
Author(s):  
Guo Hui Wu ◽  
Hong Liang Li
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Displacement Field ◽  
Line Source ◽  
Ground Surface ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Function Solution ◽  
Polar Coordinate ◽  
Inclusion Structure

In mechanical engineering and modern municipal construction, shallow-buried inclusion structure is used widely. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with double shallow-buried inclusions while bearing anti-plane harmonic line source force at any point. In complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by the circle inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition of the circle inclusions in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. Green's function, that is, the total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the circle inclusions and the location of the line source force. Based on this solution, the problem of interaction of double circular inclusions and a linear crack in semi-space can be investigated further.

Green's Function Solution of the Semi-Cylindrical Gap with Multiple Shallow-Buried Inclusions

2011 ◽  
Vol 261-263 ◽  
pp. 863-867 ◽  
Author(s):  
Hong Liang Li
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Earthquake Engineering ◽  
Displacement Field ◽  
Line Source ◽  
Ground Surface ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Function Solution ◽  
Cylindrical Inclusions

In mechanical engineering, earthquake engineering and modern municipal construction, semi-cylindrical gap and shallow-buried inclusion structure are used widely. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with semi-cylindrical gap and multiple shallow-buried inclusions while bearing anti-plane harmonic line source force at any point. In the complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by semi-cylindrical gap and multiple cylindrical inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition of the cylindrical inclusion in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. Green's function, that is, the total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of semi-cylindrical gap , the cylindrical inclusions and the location of the line source force. Based on this solution, the problem of interaction of semi-cylindrical gap , multiple cylindrical inclusions and a linear crack in semi-space can be investigated further.

Green's Function Solution of the Semi-Space with Double Shallow-Buried Cavities

2009 ◽  
Vol 417-418 ◽  
pp. 145-148 ◽  
Author(s):  
Hong Liang Li ◽  
Cun Zhang
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Displacement Field ◽  
Line Source ◽  
Ground Surface ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Free Condition ◽  
Function Solution ◽  
Cavity Structure

In mechanical engineering and modern municipal construction, shallow-buried cavity structure is used widely. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with double shallow-buried cavities while bearing anti-plane harmonic line source force at any point. In the complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by the circle cavities comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress-free condition of the circle cavities in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. Green's function, that is, the total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the circle cavities and the location of the line source force. Based on this solution, the problem of interaction of double circular cavities and a linear crack in semi-space can be investigated further.

Green's Function Solution of Multiple Shallow-Buried Cavities and Inclusions in Elastic Semi-Space

2013 ◽  
Vol 577-578 ◽  
pp. 109-112
Author(s):  
Hong Liang Li ◽  
Lei Liu
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Earthquake Engineering ◽  
Displacement Field ◽  
Line Source ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Near Surface ◽  
Function Solution ◽  
Cylindrical Cavities

In engineering composite materials, earthquake engineering and modern municipal construction, it can be found that there are shallow-buried cavity or inclusion structure near surface. When structure is impacted by dynamic load, scattering field will be produced because of the cavity or the inclusion, and it could cause dynamic stress concentration at the edge of the cavity or inclusion. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with multiple shallow-buried cavities and inclusions while bearing anti-plane harmonic line source force at any point. In the complex plane, considering the symmetry of SH-wave scattering, the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by multiple cylindrical cavities and inclusions are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress or displacement condition of the cylindrical cavities and inclusions in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. By solving these algebraic equations the value of the unknown coefficients can be obtained. So the total wave displacement field could be got. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical cavities, inclusions and the location of the line source force.

Interaction of Semi-Cylindrical Gap and Multiple Shallow-Buried Cavities and Inclusions

2012 ◽  
Vol 157-158 ◽  
pp. 1361-1364
Author(s):  
Hong Liang Li ◽  
Dan Sun
Keyword(s):  
Earthquake Engineering ◽  
Displacement Field ◽  
Line Source ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Displacement Condition ◽  
Polar Coordinate ◽  
Cylindrical Cavities ◽  
Inclusion Structure

In mechanical engineering, earthquake engineering and modern municipal construction, It can be found that there are shallow-buried cavity or inclusion structure everywhere. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with semi-cylindrical gap and multiple shallow-buried cavities and inclusions while bearing anti-plane harmonic line source force at any point. In the complex plane, considering the symmetry of SH-wave scattering, the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by semi-cylindrical gap and multiple cylindrical cavities and inclusions are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress or displacement condition of the cylindrical cavities and inclusions in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. By solving these algebraic equations the value of the unknown coefficients can be obtained. So the total wave displacement field could be got. By using the expressions, an example is provided to show the effect of the change of relative location of semi-cylindrical gap, the cylindrical cavities and inclusions and the location of the line source force.

Interaction of Multiple Semi-Cylindrical Gaps and a Shallow-Buried Cavity

2010 ◽  
Vol 163-167 ◽  
pp. 3910-3913
Author(s):  
Rui Zhang ◽  
Hong Liang Li
Keyword(s):  
Earthquake Engineering ◽  
Displacement Field ◽  
Cylindrical Cavity ◽  
Line Source ◽  
Ground Surface ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Free Condition ◽  
Cavity Structure ◽  
Polar Coordinate

In modern municipal construction and earthquake engineering, semi-cylindrical gap and shallow-buried cavity structure are used widely. In this paper, the solution of displacement field for elastic semi-space with multiple semi-cylindrical gaps and a shallow-buried cavity while bearing anti-plane harmonic line source force at any point is studied. In the complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by multiple semi-cylindrical gaps and a cylindrical cavity comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress-free condition of the gaps and the cylindrical cavity in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of semi-cylindrical gaps , the cylindrical cavity and the location of the line source force. Based on this solution, the problem of interaction of multiple semi-cylindrical gaps , a cylindrical cavity and a linear crack in semi-space can be investigated further.

Scattering of Anti-Plane SH-Wave by the Semi-Cylindrical Gap with Multiple Shallow-Buried Inclusions in Elastic Semi-Space

2012 ◽  
Vol 569 ◽  
pp. 78-81
Author(s):  
Hong Liang Li ◽  
Jing Guo ◽  
Li Ming Cai
Keyword(s):  
Displacement Field ◽  
Ground Surface ◽  
Analytic Solutions ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Dynamic Stress Concentration Factor ◽  
Polar Coordinate ◽  
Cylindrical Inclusions ◽  
Plane Sh Wave

Semi-cylindrical gap and Multiple circular inclusions exists widely in natural media, composite materials and modern municipal construction. The scattering field produced by semi-cylindrical gap and multiple circular inclusions determines the dynamic stress concentration factor around the gap and circular inclusions, and therefore determines whether the material is damaged or not. These problems are complicated. It is hard to obtain analytic solutions except for several simple conditions. In this paper, the solution of displacement field for elastic semi-space with semi-cylindrical gap and multiple cylindrical inclusions by anti-plane SH-wave is constructed. In complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane SH-wave and the scattering displacement field impacted by the gap and the cylindrical inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition around the edge of the gap and cylindrical inclusions. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane SH-wave and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical inclusions.

Scattering of Anti-Plane SH-Wave by Multiple Cylindrical Inclusions in Elastic Semi-Space

2012 ◽  
Vol 525-526 ◽  
pp. 305-308
Author(s):  
Hong Liang Li ◽  
Yong Yang
Keyword(s):  
Displacement Field ◽  
Ground Surface ◽  
Analytic Solutions ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Dynamic Stress Concentration Factor ◽  
Polar Coordinate ◽  
Cylindrical Inclusions ◽  
Plane Sh Wave

Multiple circular inclusions exists widely in natural media, engineering materials and modern municipal construction. The scattering field produced by multiple circular inclusions determines the dynamic stress concentration factor around the circular inclusions, and therefore determines whether the material is damaged or not. These problems are complicated, because there are many factors influenced. Researchers solved these problems by analysis and numerical methods. It is hard to obtain analytic solutions except for several simple conditions. In this paper, the solution of displacement field for elastic semi-space with multiple cylindrical inclusions by anti-plane SH-wave is constructed. In complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane SH-wave and the scattering displacement field impacted by the cylindrical inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition around the edge of cylindrical inclusions. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane SH-wave and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical inclusions. Based on this solution, the problem of interaction of multiple cylindrical inclusions and a linear crack in semi-space can be investigated further.

Scattering of Anti-Plane SH-Wave by Multiple Cylindrical Cavities in Elastic Semi-Space

2011 ◽  
Vol 317-319 ◽  
pp. 2497-2500
Author(s):  
Rui Zhang ◽  
Hong Liang Li
Keyword(s):  
Displacement Field ◽  
Ground Surface ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Free Condition ◽  
Scattering Field ◽  
Polar Coordinate ◽  
Cylindrical Cavities ◽  
Plane Sh Wave

In natural medium, engineering materials and structures, it can be found that there are cavities everywhere. When structure is impacted by dynamic load, the scattering field will be produced because of the cavities, and it could cause dynamic stress concentration at the edge of the cavities. In this paper, the solution of displacement field for elastic semi-space with multiple cylindrical cavities by anti-plane SH-wave is constructed. In complex plane, considering the symmetry of SH-wave scattering, the displacement field aroused by the anti-plane SH-wave and the scattering displacement field impacted by the cylindrical cavities comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress free condition of the cylindrical cavities in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane SH-wave and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical cavities. Based on this solution, the problem of interaction of multiple cylindrical cavities and a linear crack in semi-space can be investigated further.

The Green’S Function Solution of Right-Angle Plane Including a Semi-Cylindrical Canyon

2013 ◽  
Vol 275-277 ◽  
pp. 830-835
Author(s):  
Chun Xiang Zhao ◽  
Hui Qi
Keyword(s):  
Free Boundary ◽  
Green’S Function ◽  
Wave Field ◽  
Green's Function ◽  
Complex Function ◽  
Algebraic Equations ◽  
Function Solution ◽  
Out Of Plane ◽  
Stress Boundary Conditions ◽  
The Right

The Green’s function of a right-angle plane including semi-cylindrical canyon while bearing out-of-plane harmonic line source load on horizontal interface have been considered using the methods of complex function and image. Firstly, the wave field of right-angle plane was imaged half space, the scattering wave field, which satisfies the free stress boundary conditions of the right-angle plane on the vertical interface could be constructed. Secondly, a series of infinite algebraic equations be obtained to settle this problem by considering the stress free boundary condition of semi-cylindrical canyon. Finally, some examples for ground motion of a right-angle plane were given and discussed. Numerical results show that displacement of the horizontal surface is influenced by right-angle free boundary.

Dynamic Response of Out-Plane Line Loads by a Shallow-Embedded Circular Lining Structure

2011 ◽  
Vol 55-57 ◽  
pp. 1107-1110
Author(s):  
Ming Song Gao ◽  
Zhi Gang Chen
Keyword(s):  
Dynamic Response ◽  
Green’S Function ◽  
Green's Function ◽  
Stress Condition ◽  
Ground Surface ◽  
Arbitrary Point ◽  
Algebraic Equations ◽  
Dynamic Stress Concentration Factor ◽  
Lining Structure ◽  
Line Loads

The dynamic response problems of out-plane line loads by a shallow-embedded circular lining structure were investigated here by using the method of Green’s Function. Firstly a suitable Green’s function was constructed, which is an essential solution to the displacement field possessing a shallow-embedded circular lining structure while bearing out-plane harmonic line loads at an arbitrary point. Then we obtained a series of algebraic equations to solve this problem after constructing scattering waves that satisfied the zero-stress condition on the ground surface. Lastly, some numerical examples are given to show the effects that different parameters influence dynamic stress concentration factor (DSCF) by out-plane line source loads.

Sign in / Sign up

Export Citation Format

Share Document